Question: Seven points are evenly spaced out on a circle and connected as shown below to form a 7-pointed star.  What is the sum of the angle measurements of the seven tips of the star, in degrees?  One such angle is marked as $\alpha$ below.

[asy]
dotfactor=4;
draw(Circle((0,0),1));
real x = 2*pi/7;
pair A,B,C,D,E,F,G;
A=(cos(4*x), sin(4*x));
B=(cos(3*x), sin(3*x));
C=(cos(2*x), sin(2*x));
D=(cos(x), sin(x));
E=(cos(5*x), sin(5*x));
F=(cos(6*x), sin(6*x));
G=(cos(7*x), sin(7*x));
dot(A); dot(B); dot(C); dot(D); dot(E); dot(F); dot(G); dot((0,0));
label("$A$",A,W); label("$B$",B,W); label("$C$",C,N); label("$D$",D,N); label("$E$",G,ENE); label("$F$",F,SE); label("$G$",E,S);
draw(A--C--G--E--B--D--F--cycle); label("$\alpha$",C, - 1.5*dir(C));
[/asy]
Solution: The seven points divide the circumference of the circle into seven equal small arcs each with measure $\frac{360^\circ}{7}$.

$\angle ACE$ cuts off a minor arc $\widehat{AE}$, which consists of three small arcs and thus \[\widehat{AE}=3\cdot \frac{360^\circ}{7}.\]It follows that \[\angle ACE = 3\cdot \frac{360^\circ}{7} \cdot\frac{1}{ 2} = \frac{3\cdot 180^\circ}{7}.\]Each tip of the star is formed by an angle which cuts off three small arcs in a similar fashion.  Thus each tip of the star measures $\frac{3\cdot 180^\circ}{7}$ and hence all seven tips of the star together measure $3\cdot 180^\circ = \boxed{540}$ degrees.